---- > [!definition] Definition. ([[character table]]) > Let $\text{Irr }G$ denote the non-equivalent [[irreducible group representation|irreducible characters]] of $G$ and $\text{Cl }G$ the [[conjugate|conjugacy classes]]. Then, the square [[matrix]] $\left[\chi([c])\right]_{\chi \in \text{Irr }G, [c] \in \text{Cl }G}$ > is called the **character table** of $G$. > [!basicexample] > - [[character tables of small groups]] > - [[Artin 10.4.9]] > - [[Artin 10.4.10]] > [!basicexample] Example. (Character Table of $A_{4}$) > Let $A_{4}$ be the [[alternating group]] on $4$ letters. The [[conjugate|conjugacy classes of]] $A_{4}$ are > - $\{ e \}=[e]$; > - $\{ (12)(34), (13)(24), (14)(23) \}=[(12)(34)],$ (disjoint products of two-cycles);. > - $[(123)]$ > - $[132]$ > > so we know the table will have four rows and columns. We know $C_{2} \times C_{2}$ is a [[normal subgroup]] of $A_{4}$, and therefore $A_{4} \cong (C_{2} \times C_{2}) \rtimes C_{3}$. We can then lift the [[character tables of small groups|three one-dimensional irreducibles]] from $C_{3}$ to $A_{4}$ via the [[kernel iff normal subgroup|quotient map]] $G \to C_{3}$ [[first isomorphism theorem|whose]] [[kernel of a group homomorphism|kernel]] $K$ is $C_{2} \times C_{2}$, yielding [[irreducible group representation|irreducible]] [[group representation|representations]] of $A_{4}$ defined e.g. as $\rho(g):=\rho^{(C_{3})}(gK)$. Then the [[class function|orthogonality relations]] and the fact that $1^{2}+1^{2}+1^{2} + ?^{2} = 12$ implies $?=3$ allow us to fill in the final row. > > | $A_{4}$ | $[e]$ | $[(12)(34)]$ | $[(123)]$ | $[132]$ | > |:----------:|:----:|:--------:|:-----:|:-----:| > | $\chi_{\mathbb{1}}$ | 1 | 1 | 1 | 1 | > | $\chi_{2}$ | 1 | 1 | $\omega$ | $\omega^{2}$ | > | $\chi_{3}$| 1 | 1 | $\omega^{2}$ | $\omega$ | > | $\chi_{4}$ | 3 | -1 | 0 | 0 | > > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```